Method of generating an  integrated fuzzy-based guidance law for aerodynamic missiles

ABSTRACT

The method for generating an integrated guidance law for aerodynamic missiles uses a strength Pareto evolutionary algorithm (SPEA)-based approach for generating an integrated fuzzy guidance law, which includes three separate fuzzy controllers. Each of these fuzzy controllers is activated in a unique region of missile interception. The distribution of membership functions and the associated rules are obtained by solving a nonlinear constrained multi-objective optimization problem in which final time, energy consumption, and miss distance are treated as competing objectives. A Tabu search is utilized to build a library of initial feasible solutions for the multi-objective optimization algorithm. Additionally, a hierarchical clustering technique is utilized to provide the decision maker with a representative and manageable Pareto-optimal set without destroying the characteristics of the trade-off front. A fuzzy-based system is employed to extract the best compromise solution over the trade-off curve.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to guidance systems for surface-to-airmissiles, and particularly to a method of generating an integrated,fuzzy-based guidance law for aerodynamic missiles that uses a strengthPareto evolutionary algorithm (SPEA) based approach and a Tabu search todetermine the initial feasible solution for the algorithm to selectbetween one of three fuzzy controllers implementing different guidancelaws to issue a guidance command to the missile.

2. Description of the Related Art

Guidance technology of missiles includes many well known guidance lawswhich are regularly utilized. The guidance and control laws typicallyused in current tactical missiles are mainly based on classical controldesign techniques. These conventional control approaches, however, areoften not sufficient to obtain accurate tracking and interception of amissile. Therefore, advanced control theory must be applied to a missileguidance and control system in order to improve its performance. Fuzzycontrol has suitable properties to eliminate such difficulties, however,at the present time, there is very limited research related to fuzzymissile guidance design.

Fuzzy logic has been applied to change the gain of the proportionalnavigation guidance (PNG) law. Such a fuzzy-based controller was alsoused in the design of guidance laws where the line of sight (LOS) angleand change of LOS angle rate are used as input linguistic variables, andthe lateral acceleration command can be used as the output linguisticvariable for the fuzzy guidance scheme. It is known that these fuzzyguidance schemes perform better than traditional proportional navigationor augmented proportional navigation schemes; i.e., these methods resultin smaller miss distances and lower acceleration commands.

In the above, though, the parameters of the fuzzy guidance law aregenerated by trial and error, which consumes time and effort, as well ascomputational power, and the results are not necessarily optimal.Moreover, such conventional methods and systems use only one type ofguidance through the entire interception range. Each of the classicalguidance laws has a particular region of operation in which they arefound to be superior to other guidance laws.

In general, the multi-objective missile guidance law design problem canbe converted to a single objective problem through the linearcombination of different objectives as a weighted sum. The importantaspect of this weighted sum method is that a set of non-inferior (orPareto-optimal) solutions can be obtained by varying the weights.Unfortunately, this requires multiple runs, with the number runs beingequivalent to the number of desired Pareto-optimal solutions.Furthermore, this method cannot be used to find Pareto-optimal solutionsin problems having a non-convex Pareto-optimal front.

Evolutionary algorithms, however, may be used to efficiently eliminatemost of the difficulties of classical methods. Since they use apopulation of solutions in their search, multiple Pareto-optimalsolutions can be found in one single run. A multi-objective evolutionaryalgorithm (MOEA) must be started with a feasible solution, which isusually obtained by trial and error, thus requiring very highcomputational time.

It would be desirable to make such a methodology more efficient throughthe usage of a systematic technique to get the initial feasiblesolution, such as through the usage of a Tabu search (TS). TS is ahigher level heuristic algorithm for solving combinatorial optimizationproblems. It is an iterative improvement procedure which starts from anyinitial solution and attempts to determine a better solution. TS hasrecently become a well-established optimization approach that is rapidlyspreading to a variety of fields.

Now referring to actual missile guidance and control, we assume for thesake of simplicity that a missile's motion is constrained in thevertical plane. Furthermore, the missile may be modeled as a point masswith aerodynamic forces applied at the center of gravity. Thus, from themissile's balanced forces shown in FIG. 2, the equations of motion forthe missile can be written as:

$\begin{matrix}{{\overset{.}{\gamma}}_{m} = {\frac{\left( {L + {T\; \sin \; \alpha}} \right)}{m\; V_{m}} - \frac{g\; \cos \; \gamma_{m}}{V_{m}}}} & \left( {1a} \right) \\{{\overset{.}{V}}_{m} = {\frac{\left( {{T\; \cos \; \alpha} - D} \right)}{m} - {g\; \sin \; \gamma_{m}}}} & \left( {1b} \right) \\{{\overset{.}{x}}_{m} = {V_{m}\cos \; \gamma_{m}}} & \left( {1c} \right) \\{{\overset{.}{h}}_{m} = {V_{m}\sin \; \gamma_{m}}} & \left( {1d} \right) \\{L = {\frac{1}{2}\rho \; V_{m}^{2}S_{ref}C_{L}}} & \left( {1e} \right) \\{C_{L} = {C_{L\; \alpha}\left( {\alpha - \alpha_{o}} \right)}} & \left( {1f} \right) \\{D = {\frac{1}{2}\rho \; V_{m}^{2}S_{ref}C_{D}}} & \left( {1g} \right) \\{C_{D} = {C_{Do} + {k\; C_{L}^{2}}}} & \left( {1h} \right)\end{matrix}$

where L, D, and T represent the lift, drag and thrust forces acting onthe missile, respectively, ρ is the air density, S_(ref) is thereference surface area, Y_(m) represents the missile heading angle, αrepresents the missile angle of attack, m represents the missile mass,V_(m) represents the missile velocity, g is the gravitationalacceleration, x_(m) and h_(m) are the horizontal and vertical positionsof the missile, respectively, C_(L) represents the lift coefficient, andC_(D) represents the drag coefficient.

The aerodynamic derivatives C_(Lα), C_(D0) and k are given as functionsof the Mach number M, while the thrust and the mass are functions oftime. The angle of attack α is used as the control variable and themissile normal acceleration can be determined from:

$\begin{matrix}{a_{m} = {{{\overset{.}{\gamma}}_{m}V_{m}} = {\frac{\left( {L + {T\; \sin \; \alpha}} \right)}{m} - {g\; \cos \; \gamma_{m}}}}} & (2)\end{matrix}$

where the target is assumed to be a point mass with a constant velocityV_(t) and acceleration a_(t). The direction and position of the targetin the horizontal and vertical directions are determined from thefollowing relations:

$\begin{matrix}{{\overset{.}{\gamma}}_{t} = \frac{a_{t}}{V_{t}}} & \left( {3a} \right) \\{{\overset{.}{x}}_{t} = {V_{t}\cos \; \gamma_{t}}} & \left( {3b} \right) \\{{\overset{.}{h}}_{t} = {V_{t}\sin \; {\gamma_{t}.}}} & \left( {3c} \right)\end{matrix}$

From the interception geometry shown in FIG. 3, the line of sight anglerate and the derivative of the relative distance between the missile andthe target can be written as:

{dot over (θ)}=(V _(m) sin (θ−γ_(m))−V _(t) sin (θ−γ_(t)))/r  (4a)

{dot over (r)}=−V _(m) cos (θ−γ_(m))+V _(t) cos (θ−γ_(t)).  (4b)

In the above, θ represents the line of sight angle, r represents thedistance between the missile and the target, and V_(t) represents thetarget velocity. For any surface-to-air missile, there are threeguidance phases. The first phase of the trajectory is called the“launch” or “boost” phase, which occurs for a relatively short time. Thefunction of the launch phase is to take the missile away from thelauncher base. At the completion of this phase, midcourse guidance isinitiated. The function of the midcourse guidance phase is to bring themissile near to the target in a short time. The last few seconds of theengagement constitute the terminal guidance phase, which is the mostcrucial phase, since its success or failure determines the success orfailure of the entire mission.

There are two basic guidance laws governing homing missiles: PursuitGuidance (PG) and the Proportional Navigation Guidance (PNG). PG guidesthe missile to the current position of the target, whereas PNGorientates the missile to an estimated interception point. Therefore,PNG has smaller interception time than PG, but this method may showunstable behavior for excessive values of the navigation constant. Thus,it is recommended to use PNG in the launching phase in order to get thefastest heading to the target, since stability is not a relatively largeproblem in this stage, while using PG in the terminal phase.

Since PNG is used during the boost phase to direct the missile velocityto the predicted interception location, the missile velocity should bealigned with the predicted interception velocity. Therefore, the missilecommand should be a function of a velocity error angle σ and itsderivative. In the terminal phase, the position error dominates thefinal miss distance, thus it is recommended to use PG. The missilecommand must thusly be a function of the heading error δ in order tohave a stable system with a minimum miss distance. During the midcoursephase, it is hoped that the missile reaches the terminal phase with thehighest speed for the greatest distance possible and, at the same time,with a minimal heading error. Thus, the missile acceleration is afunction of both variables.

The estimated value of the angle of this direction γ_(p), can beobtained directly from the interception geometry in FIG. 3 as:

$\begin{matrix}{\gamma_{p} = {\theta - {{{atan}\left( \frac{V_{T}t_{p}{\sin (\varphi)}}{r + {V_{T}t_{p}{\cos (\varphi)}}}\; \right)}.}}} & (5)\end{matrix}$

The derivative of this angle is:

$\begin{matrix}{{\overset{.}{\gamma}}_{p} = {\overset{.}{\theta} - \frac{V_{T}{t_{p}\left\lbrack {{{- \overset{.}{r}}\; \sin \; \phi} + {\overset{.}{\phi}\left( {{V_{T}t_{p}} + {r\; \cos \; \phi}} \right)}} \right\rbrack}}{\left( {V_{T}t_{p}} \right)^{2} + {2r\; V_{T}t_{p}\cos \; \phi} + r^{2}}}} & (6)\end{matrix}$

where t_(p) is the predicted time to intercept the target, which can besimply estimated as:

$\begin{matrix}{t_{p} \approx {- {\frac{r}{\overset{.}{r}}.}}} & (7)\end{matrix}$

As will be discussed in greater detail below, a is a variable definingthe distribution of the membership function. It would be desirable touse a multi-objective evolutionary algorithm (MOEA) in order to generatemissile guidance laws without requiring start points found via trial anderror. Thus, a method of generating an integrated guidance law foraerodynamic missiles solving the aforementioned problems is desired.

SUMMARY OF THE INVENTION

The present invention relates to the generation of an integratedguidance law for aerodynamic missiles. Particularly, a strength Paretoevolutionary algorithm (SPEA)-based approach is utilized for generatingan integrated fuzzy guidance law, which includes three separate fuzzycontrollers. Each of these fuzzy controllers is activated in a uniqueregion of missile interception.

The distribution of membership functions and the associated rules areobtained by solving a nonlinear constrained multi-objective optimizationproblem in which final time, energy consumption, and miss distance aretreated as competing objectives. Further, a Tabu search is utilized tobuild a library of initial feasible solutions for the multi-objectiveoptimization algorithm.

Additionally, a hierarchical clustering technique is utilized to providethe decision maker with a representative and manageable Pareto-optimalset without destroying the characteristics of the trade-off front. Afuzzy-based system is employed to extract the best compromise solutionover the trade-off curve.

The method includes the following steps: (a) establishing a missilelaunch guidance law f₁(z), a missile midcourse guidance law f₂(z) and amissile terminal guidance law f₃(z), wherein z represents a vectorcontaining fuzzy membership functions and guidance rules associated witheach of the missile guidance laws; and (b) optimizing the missile launchguidance law f₁(z), the missile midcourse guidance law f₂(z) and themissile terminal guidance law f₃(z) by simultaneously minimizing thefollowing set of equations:

${Minmize}\mspace{14mu} \left\{ \begin{matrix}{{f_{1}(z)} = t_{f}} \\{{f_{2}(z)} = {\int_{0}^{t_{f}}{a_{M}^{2}{t}}}} \\{{f_{3} = {r\left( t_{f} \right)}},}\end{matrix} \right.$

wherein t_(f) represents missile interception time, t represents time, rrepresents a distance between the missile and a target, and a_(M)represents a missile normal acceleration, and further|r(t_(f))|<r_(miss-allowed), where r_(miss-allowed) representspre-selected allowable miss distance.

The minimization is performed by a strength Pareto evolutionaryalgorithm having the following steps: (c) initializing a feasiblepopulation by generating an initial population and generating an emptyexternal Pareto-optimal set, where the feasible population is selectedto satisfy a missile guidance constraint, wherein the missile guidanceconstraint is the pre-selected allowable miss distance; (d) searchingthe feasible population for non-dominated individuals and copying thenon-dominated individuals into the external Pareto set; (e) searchingthe external Pareto set for the non-dominated individuals and removingall dominated solutions from the external Pareto set.

Then, (f) if the number of the individuals stored in the external Paretoset exceeds a pre-specified maximum size, then reducing the set byclustering; (g) assigning a strength to each individual in the externalPareto set, where the strength is proportional to the number ofindividuals covered by that individual; (h) calculating a fitness ofeach individual in the population as the sum of the strengths of allexternal Pareto solutions which dominate that individual. A smallpositive number is added to the resulting sum to guarantee that Paretosolutions are most likely to be produced;

Then, the method proceeds by: (i) combining the population and theindividuals of the external Pareto set; (j) selecting two individuals atrandom and comparing their respective fitnesses; (k) selecting theindividual with the greater fitness and copying the individual with thegreater fitness to a mating pool; (l) performing crossover and mutationoperations to generate a new population.

Finally, the method concludes by: (m) checking for pre-selected stoppingcriteria, where the pre-selected stopping criteria includes a missdistance that is less than the pre-selected allowable miss distance. Ifa pre-selected stopping criterion is satisfied, then optimization isceased and the optimal population is recorded. If a pre-selectedstopping criterion is not satisfied, then the previous population isreplaced with the new population and the method returns to step (d).

In the above, the search is stopped if the generation counter exceedsits maximum number.

These and other features of the present invention will become readilyapparent upon further review of the following specification anddrawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a system implementing the method ofgenerating an integrated guidance law for aerodynamic missiles accordingto the present invention.

FIG. 2 is a diagram of a missile illustrating the variables typicallyutilized in aerodynamic missile guidance laws.

FIG. 3 is a diagram illustrating typical interception geometry foraerodynamic missiles.

FIG. 4 is a chart illustrating typical normalized membership functionsutilized in the method of generating an integrated guidance law foraerodynamic missiles according to the present invention.

FIG. 5 is a chart illustrating a typical structure of a geneticalgorithm individual.

FIG. 6 is a graph illustrating an exemplary time history of missile massand thrust.

FIG. 7 is a flowchart illustrating the steps of a multi-objectiveevolutionary algorithm used in the method of generating an integratedguidance law for aerodynamic missiles according to the presentinvention.

FIG. 8A is a Pareto front graph illustrating an exemplary missile'sconsumed energy as a function of time.

FIG. 8B is a Pareto front graph illustrating an exemplary missile'stravel distance as a function of time.

FIG. 9 is a comparison graph illustrating exemplary time history of theinterception of a maneuvering target.

FIG. 10 is a comparison graph illustrating exemplary time history of anangle of attack for the maneuvering target of FIG. 9.

FIGS. 11A, 11B and 11C are graphs illustrating optimal membershipfunctions for launching controllers associated with the multi-objectiveevolutionary algorithm of FIG. 7.

FIGS. 12A, 12B and 12C are graphs illustrating optimal membershipfunctions for midcourse controllers associated with the multi-objectiveevolutionary algorithm of FIG. 7.

FIGS. 13A, 13B and 13C are graphs illustrating optimal membershipfunctions for terminal controllers associated with the multi-objectiveevolutionary algorithm of FIG. 7.

FIG. 14 is a comparison graph illustrating time history of controlaction generated by the integrated guidance law for aerodynamic missilesaccording to the present invention with differing signal-to-noiseratios.

FIG. 15 is a comparison graph illustrating interception history for anexemplary non-maneuvering target.

FIG. 16 is a block diagram illustrating system components forimplementing the method of generating an integrated guidance law foraerodynamic missiles according to the present invention.

These and other features of the present invention will become readilyapparent upon further review of the following specification anddrawings.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

FIG. 1 diagrammatically illustrates an overview of the integratedguidance law generated by the present method of generating an integratedguidance law for aerodynamic missiles. The method utilizes threeseparate fuzzy-based guidance laws for the launch, mid-course andterminal phases, respectively. In FIG. 1, the launch fuzzy lawcontroller (FLC) is shown as block 12, the midcourse FLC is shown asblock 14, and the terminal FLC is shown as block 16. A fuzzy switchingcontroller 10, with two gains, is used to provide smooth transitioningbetween the three separate guidance laws 12, 14, 16. These switchinggains are determined from the following fuzzy rules:

If r is Big (B) then K ₁=1 and K ₂=0 (launch phase);  (8a)

If r is Medium (M) then K ₁=0 and K ₂=0 (Midcourse phase);  (8b)

If r is Small (S) then K ₁=0 and K ₂=1 (Terminal Phase).  (8c)

The fuzzy controller 10 has three main components: scaling factors,membership functions and rules. The initial point in the generation ofthe fuzzy guidance law is to choose numbers and shapes of membershipfunctions (MFs) for input and output variables. In the following, MFswith triangular shapes are chosen for all input and output variables, asillustrated in FIG. 4. All of the variables have positive and negativevalues, except for the range which is solely positive. Thus, only threeMFs are used for the range and five MFs are used for the othervariables.

The second step in the generation is to determine the scaling factorswhich convert the physical ranges of the fuzzy variables into thenormalized ranges between −1 and 1. The scaling factors can bedetermined from the expected maximum values of the controller'svariables, which are typically obtained from the engineering dataregarding the particular missile's dynamics.

in order to complete the definition of the fuzzy guidance law, the ruleswhich define the relationship between the control action andmissile-target measurements should be determined. To include thelinguistic rules in the optimization process, an integer encoding systemis used to refer to the output fuzzy variables, as shown below in Table1:

TABLE I Encoding System for the FLC Output MF NB NS ZO PS PB Code 1 2 34 5

In FIG. 1, the launch FLC guidance law 12 is shown as a function ƒ(σ,{dot over (σ)}) or, in other words, as a function of the velocity errorangle and its time derivative. The midcourse guidance law 14 is shown asa function ƒ(σ, δ), or as a function of the error angle and the headingerror. Missile and target measurements are used as an input for θ(σ, δ).Similarly, the terminal FLC guidance law 16 is given as a function ƒ(δ,{dot over (δ)}) or, in other words, as a function of the heading errorand its time derivative.

The two gains of the fuzzy switching system 10 K₁, K₂ are, respectively,the output of launch FLC guidance law 12 and terminal FLC guidance law16 (following output paths u₁ and u₃ to blocks 18, 22, respectively, inFIG. 1). The output of the midcourse FLC guidance law 14 (following pathu₂ to block 20 in FIG. 1) is a function of K₁ and K₂, namely,(1−K₁)(1−K₂). The input to the fuzzy switching controller 10 is thedistance between the missile and the target r, and the ultimate output uis the missile guidance command.

In order to optimally tune the fuzzy parameters, the optimizationproblem can be formulated as follows:

$\begin{matrix}{{Minmize}\mspace{14mu} \left\{ \begin{matrix}{{f_{1}(z)} = t_{f}} \\{{f_{2}(z)} = {\int_{0}^{t_{f}}{a_{M}^{2}{t}}}} \\{f_{3} = {r\left( t_{f} \right)}}\end{matrix} \right.} & (9)\end{matrix}$

which is the simultaneous minimization of the three fuzzy guidance lawsf₁ 12, f₂ 14 and f₃ 16, respectively, subject to the condition|r(t_(f))|<r_(miss-allowed). In the above, t_(f) is the interceptiontime, r_(miss-allowed) is the allowed miss distance, and z is a vectorthat contains the unknown parameters of the fuzzy guidance law (i.e.,MFs and rules), as illustrated in FIG. 5. The rules are denoted byr_(n), which can take only integer numbers between 1 and 5, according tothe code shown in Table 1. The variables a that define the distributionof the membership function are real, with a range between 0 and 1.0, asshown in FIG. 4. In this problem, 45 rules and 20 variables define theMFs distributions.

In practice, many real-world problems involve simultaneous optimizationof several objective functions. Generally, these functions arenon-commensurable and often conflicting objectives. Multi-objectiveoptimization with such conflicting objective functions gives rise to aset of optimal solutions, rather than only one optimal solution. Thereason for the optimality of many solutions is that no single solutioncan be considered to be better than any other with respect to allobjective functions. These optimal solutions are generally known as“Pareto-optimal” solutions.

A general multi-objective optimization problem consists of a number ofobjectives to be optimized simultaneously, and is associated with anumber of equality and inequality constraints. The generalmulti-objective optimization problem can be formulated as follows:

$\begin{matrix}{{{{Minimize}\mspace{14mu} {f_{i}(x)}\mspace{14mu} i} = 1},\ldots \mspace{14mu},N_{obj}} & (10) \\{{Subject}\mspace{14mu} {to}\text{:}\mspace{14mu} \left\{ \begin{matrix}{{g_{j}(x)} = 0} & {{j = 1},\ldots \mspace{14mu},M,} \\{{h_{k}(x)} \leq 0} & {{k = 1},\ldots \mspace{14mu},K,}\end{matrix} \right.} & (11)\end{matrix}$

where f_(i) is the i^(th) objective function, x is a decision vectorthat represents a solution, and N_(obj) is the number of objectives.

For a multi-objective optimization problem, any two solutions x₁ and x₂can have one of two possibilities: either one dominates the other ornone are dominant. In a minimization problem, without loss ofgenerality, a solution x₁ dominates x₂ if and only if the following twoconditions are satisfied:

∀iε{1, 2, . . . , N _(obj) }:f _(i)(x ₁)≦f _(i)(x ₂)  (12)

∃j{1, 2, . . . , N _(obj) }:f _(j)(x ₁)<f _(f)(x ₂)  (13)

If either of the above conditions is violated, the solution x₁ does notdominate the solution x₂. If x₁ dominates the solution x₂, then x₁ iscalled the “non-dominated solution” within the set. The solutions thatare non-dominated within the entire search space are denoted asPareto-optimal and constitute the Pareto-optimal set or Pareto-optimalfront.

The strength Pareto evolutionary algorithm (SPEA) is an algorithmspecifically designed for multi-objective optimization. This techniquestores externally the individuals that represent a non-dominated frontamong all solutions considered thus far. All individuals in the externalset participate in selection. SPEA uses the concept of Pareto dominancein order to assign scalar fitness values to individuals in the currentpopulation. The algorithm begins with assignment of a real value s in[0,1] (called the “strength”) to each individual in the Pareto-optimalset. The strength of an individual is proportional to the number ofindividuals covered by it. The strength of a Pareto solution is, at thesame time, its fitness.

Subsequently, the fitness of each individual in the population is thesum of the strengths of all external Pareto solutions by which it iscovered. In order to guarantee that Pareto solutions are most likely tobe produced, one is added to the resulting value. This fitnessassignment ensures that the search is directed towards the non-dominatedsolutions and, at the same time, the diversity among dominated andnon-dominated solutions is maintained.

Generally, the algorithm includes the following steps. First,Initialization: Generate an initial population and create the emptyexternal Pareto-optimal set. Second, External set updating: The externalPareto-optimal set is updated by: (a) searching the population for thenon-dominated individuals and copying them to the external Pareto set;(b) searching the external Pareto set for the non-dominated individualsand removing all dominated solutions from the set; and (c) if the numberof the individuals externally stored in the Pareto set exceeds apre-specified maximum size, reducing the set by clustering.

Third, Fitness assignment: Calculate the fitness values of individualsin both the external Pareto set and the population by: (a) assigning thestrength s for each individual in the external set (the strength isproportional to the number of individuals covered by that individual);and (b) the fitness of each individual in the population is the sum ofthe strengths of all external Pareto solutions that dominate thatindividual. A small positive number is added to the resulting sum toguarantee that Pareto solutions are most likely to be produced.

Fourth, Selection: Combine the population and the external setindividuals. Select two individuals at random and compare their fitness.Select the better one and copy it to the mating pool. Fifth, Crossoverand Mutation: Perform the crossover and mutation operations according totheir probabilities to generate the new population. Sixth and finally,Termination: check for stopping criteria. If any one criterion issatisfied, then stop. Otherwise, copy new population to old populationand return to the Second step. In the following, the search will bestopped if the generation counter exceeds its maximum number.

The present method restricts the search within the feasible region.Therefore, a procedure is imposed to check the feasibility of theinitial population individuals and the generated children throughgenetic algorithm (GA) operations. This ensures feasibility of thenon-dominated solutions. However, filling the initial population withfeasible solutions is a relatively time-consuming step, particularlywith large-scale systems where the number of control variables is high.In such a case, producing a feasible solution randomly is relativelydifficult and time-consuming.

Thus, the present method builds a database of feasible solutions tobegin the SPEA technique with initial feasible solutions. Themethodology used to generate the feasible solutions is the Tabu searchtechnique. The Tabu search is a well known mathematical optimizationmethod, belonging to the class of local search techniques. Tabu searchenhances the performance of a local search method by using memorystructures; i.e., once a potential solution has been determined, it ismarked as “taboo” (“tabu” being a different spelling of the same word)so that the algorithm does not visit that possibility repeatedly.

The Pareto-optimal set can be extremely large, or even contain aninfinite number of solutions. In this case, reducing the set ofnon-dominated solutions without destroying the characteristics of thetrade-off front is desirable from the decision maker's point of view. Anaverage linkage-based hierarchical clustering algorithm is employed toreduce the Pareto set to manageable size. The algorithm worksiteratively by joining the adjacent clusters until the required numberof groups is obtained.

In the present method, fuzzy set theory is implemented to efficientlyderive a candidate Pareto-optimal solution for the decision makers. Upongeneration of the Pareto-optimal set, the method presents a fuzzy-basedmechanism to extract a Pareto-optimal solution as the best compromisesolution. Due to the generally imprecise nature of the decision maker'sjudgment, the i-th objective function of a solution in thePareto-optimal set, denoted as F_(i), is represented by a membershipfunction μ_(i), which is defined as:

$\begin{matrix}{\mu_{i} = \left\{ \begin{matrix}{1,} & {{F_{i} \leq F_{i}^{m\; i\; n}},} \\{\frac{F_{i}^{{ma}\; x} - F_{i}}{F_{i}^{{ma}\; x} - F_{i}^{m\; i\; n}},} & {{F_{i}^{m\; i\; n} < F_{i} < F_{i}^{{ma}\; x}},} \\{0,} & {F_{i} \geq {F_{i}^{m\; {ax}}.}}\end{matrix} \right.} & (14)\end{matrix}$

where F_(i) ^(max) and F_(i) ^(min) are the maximum and minimum valuesof the i-th objective function, respectively.

For each non-dominated solution k, the normalized membership functionμ^(k) is calculated as:

$\begin{matrix}{\mu^{k} = \frac{\sum\limits_{i = 1}^{N_{obj}}\mu_{i}^{k}}{\sum\limits_{j = 1}^{M}{\sum\limits_{i = 1}^{N_{obj}}\mu_{i}^{j}}}} & (15)\end{matrix}$

where M represents the number of non-dominated solutions. The bestcompromise solution is the one having the maximum of μ^(k). Arrangingall solutions in the Pareto-optimal set in descending order according totheir membership function will provide the decision maker with apriority list of non-dominated solutions.

In order to test the accuracy of the present method, an exemplarymissile, with a corresponding set of missile parameters, is introduced.For purposes of modeling and simulation, it is assumed that the missileunder consideration has thrust and mass that vary with time, as shown inFIG. 5, while the other parameters are given as:

C _(Lα)=2.9+0.3 M+0.25 M²+0.01 M³, α₀=0

C _(Do)=0.45−0.01 M, k=0.06, S _(ref)=0.08  (16)

and the allowed miss distance is set to 2.0 m. The initial values forthe missile variables are:

v _(m)=10m/s,γ _(m)=30°,r=5000m.  (17)

The maximum allowed ranges for the fuzzy input and output variables canbe estimated as:

$\begin{matrix}{{{\alpha_{{ma}\; x} = {20{^\circ}}},{\delta_{{ma}\; x} = {\sigma_{{ma}\; x} = {20{^\circ}}}}}{{\overset{.}{\delta}}_{{ma}\; x} = {{\overset{.}{\sigma}}_{{ma}\; x} = {\frac{a_{{ma}\; x}}{600} \approx {28.6\mspace{14mu} \deg \text{/}\sec}}}}} & (18)\end{matrix}$

and the target is assumed to have a constant speed of 400 m/sec with aconstant acceleration of 3G (G=9.8 m/sec²). The initial values for themissile and target variables are:

v _(m)=10m/s,γ _(m)=30°,r=5000m

θ=50°,γ_(t)=0  (19)

and the simulation is performed using a variable step solver. Thesimulation stops when the closing velocity becomes positive. The timeand the relative distance at that instant are the final interceptiontime and the miss distance, respectively.

FIG. 7 is a flowchart of the present method. The method begins at step200. The algorithms must be started with a feasible population thatsatisfies the miss distance constraint. This initial population can beobtained randomly, but such a random generation would consume a greatdeal of time and computational power. Thus, at step 202, a Tabu searchis utilized to obtain the initial feasible solution for MOEA. At step202, the initial generation gen is set to zero. The Tabu search runs fora number of times equal to the number of individuals in each generation.At each run, the Tabu search randomly chooses a solution and tries tosearch around this solution to minimize the miss distance alone.

When the algorithm finds a solution that gives a miss distance that isless than the allowed value, it terminates and this solution isrecorded. The procedure is repeated for a number of times equal to thenumber of individuals at each generation (denoted as N_(ind)). Theobtained feasible solution is used as the initial generation of theMOEA. For the example given above, the Pareto front obtained afternearly 500 generations is shown in FIGS. 8A and 8B. A conflict betweenthe missile acceleration commands and the interception time is observed,while the miss distance increases with the increase of the interceptiontime.

If a guidance law that intercepts the target with a miss distance lessthan 10 m is chosen, then a set of controllers is produced. The bestsolution is obtained by the fuzzy algorithm. The time history of theinterception variables is shown in FIGS. 9 and 10, which indicate thatthe obtained guidance law intercepts the target successfully. Thus,without any prior knowledge regarding the guidance rules or thedistribution of the membership functions, the algorithms are able togenerate guidance laws with satisfactory performance. The correspondingrules and MFs for this guidance law are show in FIGS. 11A, 11B and 11C,FIGS. 12A, 12B and 12C, FIGS. 13A, 13B and 13C, and Tables 2, 3 and 4,given below:

TABLE 2 Best rules for the launching controller obtained from MOEA u ė eNB NS ZO PS PB NB NS NB PB NS PB NS PS PB PB PB NS ZO PS NS ZO PS NS PSPS NB NB NB NS PB NB PS NB PB PS

TABLE 3 Best rules for the terminal controller obtained from MOEA u ė eNB NS ZO PS PB NB PB NS NS ZO PB NS NS PB PB NS PS ZO PB NS ZO PS NB PSNS PS NB NB PS PB NB ZO PS PS NB

TABLE 4 Best rules for midcourse controller obtained from MOEA u ė e NBNS ZO PS PB NB NS NS NB NB ZO NS NS PB PS ZO PS ZO NS NB ZO PB PS PS NSZO NS NB PS PB ZO PB PB PS PS

FIG. 9 illustrates the performance of the integrated fuzzy guidance lawwhen the classical PD-Fuzzy rules are used with equally distributedmembership functions (i.e., all a=0.5). This case represents a guidancelaw that is generated by engineering experience alone. The angles ofattack from both guidance laws are shown in FIG. 10. It can be observedthat the final interception time is slightly higher in the optimizedcase, but the miss distance recorded for the unoptimized case isapproximately 70 cm, compared with approximately 2 cm for the optimizedcase. These results are expected, since more emphasis is placed on themiss distance the final guidance law is chosen among the set of laws inthe Pareto set. However, a low level of the required angle of attack isobserved for the optimized guidance law, which can be considered asanother advantage of the law.

Returning to FIG. 7, the present method can be described with thefollowing steps: (a) establishing a missile launch guidance law f₁(z), amissile midcourse guidance law f₂(z) and a missile terminal guidance lawf₃(z), wherein z represents a vector containing fuzzy membershipfunctions and guidance rules associated with each of the missileguidance laws; and (b) optimizing the missile launch guidance law f₁(z),the missile midcourse guidance law f₂(z) and the missile terminalguidance law f₃(Z) by simultaneously minimizing the following set ofequations:

${Minmize}\mspace{14mu} \left\{ \begin{matrix}{{f_{1}(z)} = t_{f}} \\{{f_{2}(z)} = {\int_{0}^{t_{f}}{a_{M}^{2}{t}}}} \\{{f_{3} = {r\left( t_{f} \right)}},}\end{matrix} \right.$

wherein t_(f) represents missile interception time, t represents time, rrepresents a distance between the missile and a target, and a_(M)represents a missile normal acceleration, and further|r(t_(f))|<r_(miss-allowed), where r_(miss-allowed) representspre-selected allowable miss distance.

The minimization is performed by a strength Pareto evolutionaryalgorithm having the following steps: (c) initializing a feasiblepopulation by generating an initial population and generating an emptyexternal Pareto-optimal set (step 202), where the feasible population isselected to satisfy a missile guidance constraint, wherein the missileguidance constraint is the pre-selected allowable miss distance; (d)searching the feasible population for non-dominated individuals andcopying the non-dominated individuals into the external Pareto set (step204); (e) searching the external Pareto set for the non-dominatedindividuals and removing all dominated solutions from the externalPareto set (steps 206 and 208).

Then, (f) if the number of the individuals stored in the external Paretoset exceeds a pre-specified maximum size (step 210), then reducing theset by clustering (step 214); (g) assigning a strength to eachindividual in the external Pareto set, where the strength isproportional to the number of individuals covered by that individual;(h) calculating a fitness of each individual in the population as thesum of the strengths of all external Pareto solutions which dominatethat individual. A small positive number is added to the resulting sumto guarantee that Pareto solutions are most likely to be produced.

Then, the method continues with (i) combining the population and theindividuals of the external Pareto set (step 212); (j) selecting twoindividuals at random and comparing their respective fitnesses; (k)selecting the individual with the greater fitness and copying theindividual with the greater fitness to a mating pool (step 216); (l)performing crossover and mutation operations to generate a newpopulation (step 218).

Finally, the method concludes by: (m) checking for pre-selected stoppingcriteria (step 220), where the pre-selected stopping criteria includes amiss distance that is less than the pre-selected allowable missdistance. If a pre-selected stopping criterion is satisfied, thenceasing optimization (steps 228 and 230) and recording the optimalpopulation. If a pre-selected stopping criterion is not satisfied, thenreplacing the previous population with the new population and returningto step (d) (steps 224 and 226).

In the above, the search is stopped if the generation counter exceedsits maximum number (step 222). Target position measurement is typicallynot precise and has a fuzzy distribution due to thermal and radarnoises. Thus, white noise is embedded in the measured signals, which canbe modeled by a Gaussian density function defined as:

$\begin{matrix}{{{f(x)} = {\frac{1}{\sqrt{2\pi \; \sigma^{2}}}^{- \frac{{({x - \mu})}^{2}}{2\sigma^{2}}}}},} & (20)\end{matrix}$

where μ is the mean value and σ is the standard deviation.

The Matlab function “awng” (add white noise) has been used to add noiseto three signals, which were used in evaluating the control action fromthe three fuzzy controllers. These signals are the range r, the line ofsight angle θ and the rate of change of the line of sight angle {dotover (θ)}. The performance of the best designed integrated Pareto fuzzyguidance law (IPFGL) obtained from the MOEA above with different signalto noise ratios is shown in Table 5 below:

TABLE 5 Performance of IPFGL designed by MOEA with the existence ofnoisy measurements Case Interception time, t_(f), (sec) Miss distance,(m) Acceleration 10⁻⁴ × ∫₀^(t_(f))a_(m)² t Without noise 17.03 0.02 29.64 SNR = 100 17.72 0.878 49.12 SNR = 50  18.38 6.080 88.97 SNR = 20 14.5  2482     23.2 

The results of Table 5 show that the designed IPFGLs perform well withthe existence of noisy measurements, as indicated by the small missdistance when SNR=50. When the noise becomes very high (i.e., SNR=20),the missile cannot intercept the target due to incorrect informationbeing sent to the guidance law which, in turn, sends an incorrectguidance action to the missile, as shown in FIG. 9.

It should be understood that the calculations may be performed by anysuitable computer system, such as that diagrammatically shown in FIG.16. Data is entered into system 100 via any suitable type of userinterface 116, and may be stored in memory 112, which may be anysuitable type of computer readable and programmable memory. Calculationsare performed by processor 114, which may be any suitable type ofcomputer processor and may be displayed to the user on display 118,which may be any suitable type of computer display.

Processor 114 may be associated with, or incorporated into, any suitabletype of computing device, for example, a personal computer or aprogrammable logic controller. The display 118, the processor 114, thememory 112 and any associated computer readable recording media are incommunication with one another by any suitable type of data bus, as iswell known in the art.

Examples of computer-readable recording media include a magneticrecording apparatus, an optical disk, a magneto-optical disk, and/or asemiconductor memory (for example, RAM, ROM, etc.). Examples of magneticrecording apparatus that may be used in addition to memory 112, or inplace of memory 112, include a hard disk device (HDD), a flexible disk(FD), and a magnetic tape (MT). Examples of the optical disk include aDVD (Digital Versatile Disc), a DVD-RAM, a CD-ROM (Compact Disc-ReadOnly Memory), and a CD-R (Recordable)/RW.

It is to be understood that the present invention is not limited to theembodiments described above, but encompasses any and all embodimentswithin the scope of the following claims.

1. A computerized method of generating an integrated guidance law foraerodynamic missiles, comprising the steps of: (a) establishing amissile launch guidance law f₁(z), a missile midcourse guidance lawf₂(z) and a missile terminal guidance law f₃(z), wherein z represents avector containing fuzzy membership functions and guidance rulesassociated with each of the missile guidance laws; (b) optimizing themissile launch guidance law f₁(z), the missile midcourse guidance lawf₂(z) and the missile terminal guidance law f₃(z) by simultaneouslyminimizing the following set of equations:${Minmize}\mspace{14mu} \left\{ \begin{matrix}{{f_{1}(z)} = t_{f}} \\{{f_{2}(z)} = {\int_{0}^{t_{f}}{a_{M}^{2}{t}}}} \\{{f_{3} = {r\left( t_{f} \right)}},}\end{matrix} \right.$ wherein t_(f) represents missile interceptiontime, t represents time, r represents a distance between the missile anda target, and a_(M) represents a missile normal acceleration, andfurther |r(t_(f))<r_(miss-allowed), where r_(miss-allowed) represents apre-selected allowable miss distance; wherein the minimization isperformed by a strength Pareto evolutionary algorithm having thefollowing steps: (c) initializing a feasible population by generating aninitial population and generating an empty external Pareto-optimal set,the feasible population being selected to satisfy a set of missileguidance constraints, wherein the set of missile guidance constraintsincludes the pre-selected allowable miss distance; (d) searching thefeasible population for non-dominated individuals and copying thenon-dominated individuals into the external Pareto set; (e) searchingthe external Pareto set for the non-dominated individuals and removingall dominated solutions from the external Pareto set; (f) if the numberof the individuals stored in the external Pareto set exceeds apre-specified maximum size, then reducing the set by clustering; (g)assigning a strength to each individual in the external Pareto set,wherein the strength is proportional to the number of individualscovered by that individual; (h) calculating a fitness of each individualin the population as the sum of the strengths of all external Paretosolutions which dominate that individual; (i) combining the populationand the individuals of the external Pareto set; (j) randomly selectingtwo individuals and comparing their respective fitnesses; (k) selectingthe individual with the greater fitness and copying the individual withthe greater fitness to a mating pool; (l) performing crossover andmutation operations to generate a new population; and (m) checking forpre-selected stopping criteria and if a pre-selected stopping criterionis satisfied, then ceasing optimization and recording the optimalpopulation, and if a pre-selected stopping criterion is not satisfied,then replacing the previous population with the new population andreturning to step d), wherein the pre-selected stopping criteriaincludes a miss distance that is less than the pre-selected allowablemiss distance.
 2. The computerized method of generating an integratedguidance law for aerodynamic missiles as recited in claim 1, whereinfollowing step (h), a pre-selected positive number is added to the sumof the strengths.
 3. The computerized method of generating an integratedguidance law for aerodynamic missiles as recited in claim 2, wherein theset of missile guidance constraints further includes missile final time.4. The computerized method of generating an integrated guidance law foraerodynamic missiles as recited in claim 3, wherein the set of missileguidance constraints further includes missile energy consumption.
 5. Thecomputerized method of generating an integrated guidance law foraerodynamic missiles as recited in claim 4, wherein an i-th membershipfunction μ_(i) of an i-th objective function of the Pareto-optimal setF_(i), wherein i is an integer, is defined as:$\mu_{i} = \left\{ \begin{matrix}{1,} & {{F_{i} \leq F_{i}^{m\; i\; n}},} \\{\frac{F_{i}^{{ma}\; x} - F_{i}}{F_{i}^{{ma}\; x} - F_{i}^{m\; i\; n}},} & {{F_{i}^{m\; i\; n} < F_{i} < F_{i}^{{ma}\; x}},} \\{0,} & {F_{i} \geq {F_{i}^{{ma}\; x}.}}\end{matrix} \right.$ wherein F_(i) ^(max) and F_(i) ^(min) representthe maximum and minimum values of the i-th objective function,respectively.
 6. The computerized method of generating an integratedguidance law for aerodynamic missiles as recited in claim 5, wherein foreach non-dominated solution k, a normalized membership function μ_(k) iscalculated as:${\mu^{k} = \frac{\sum\limits_{i = 1}^{N_{obj}}\mu_{i}^{k}}{\sum\limits_{j = 1}^{M}{\sum\limits_{i = 1}^{N_{obj}}\mu_{i}^{j}}}},$wherein j is an integer, N_(obj) represents the number of objectives inthe multi-objective optimization, and M represents the number ofnon-dominated solutions.
 7. A system for generating an integratedguidance law for aerodynamic missiles, comprising: a processor; computerreadable memory coupled to the processor; a user interface coupled tothe processor; software stored in the memory and executable by theprocessor, the software having: means for establishing a missile launchguidance law f₁(z), a missile midcourse guidance law f₂(z) and a missileterminal guidance law f₃(z), wherein z represents a vector containingfuzzy membership functions and guidance rules associated with each ofthe missile guidance laws; means for optimizing the missile launchguidance law f₁(z), the missile midcourse guidance law f₂(z) and themissile terminal guidance law f₃(z) by simultaneously minimizing thefollowing set of equations:${Minmize}\mspace{14mu} \left\{ \begin{matrix}{{f_{1}(z)} = t_{f}} \\{{f_{2}(z)} = {\int_{0}^{t_{f}}{a_{M}^{2}{t}}}} \\{{f_{3} = {r\left( t_{f} \right)}},}\end{matrix} \right.$ wherein t_(f) represents missile interceptiontime, t represents time, r represents a distance between the missile anda target, and a_(M) represents a missile normal acceleration, andfurther |r(t_(f))|<r_(miss-allowed), where r_(miss-allowed) represents apre-selected allowable miss distance, wherein the minimization isperformed by said means for optimizing the missile launch guidance lawimplementing a strength Pareto evolutionary algorithm, said means foroptimizing the missile launch guidance law comprising: meansinitializing a feasible population by generating an initial populationand generating an empty external Pareto-optimal set, the feasiblepopulation being selected to satisfy a set of missile guidanceconstraints, wherein the set of missile guidance constraints includesthe pre-selected allowable miss distance; means for searching thefeasible population for non-dominated individuals and copying thenon-dominated individuals into the external Pareto set; means forsearching the external Pareto set for the non-dominated individuals andremoving all dominated solutions from the external Pareto set, whereinif the number of the individuals stored in the external Pareto setexceeds a pre-specified maximum size, then the set is reduced byclustering; means for assigning a strength to each individual in theexternal Pareto set, wherein the strength is proportional to the numberof individuals covered by that individual; means for calculating afitness of each individual in the population as the sum of the strengthsof all external Pareto solutions which dominate that individual; meansfor combining the population and the individuals of the external Paretoset; means for randomly selecting two individuals and comparing theirrespective fitnesses; means for selecting the individual with thegreater fitness and copying the individual with the greater fitness to amating pool; means for performing crossover and mutation operations togenerate a new population; and means for checking for pre-selectedstopping criteria and if a pre-selected stopping criterion is satisfied,then ceasing optimization and recording the optimal population, and if apre-selected stopping criterion is not satisfied, then replacing theprevious population with the new population, wherein the pre-selectedstopping criteria includes a miss distance that is less than thepre-selected allowable miss distance.
 8. The system for generating anintegrated guidance law for aerodynamic missiles as recited in claim 7,wherein the set of missile guidance constraints further includes missilefinal time.
 9. The system for generating an integrated guidance law foraerodynamic missiles as recited in claim 8, wherein the set of missileguidance constraints further includes missile energy consumption.
 10. Acomputer software product that includes a medium readable by aprocessor, the medium having stored thereon a set of instructions forgenerating an integrated guidance law for aerodynamic missiles, theinstructions comprising: a) a first sequence of instructions which, whenexecuted by the processor, causes the processor to establish a missilelaunch guidance law f₁(z), a missile midcourse guidance law f₂(z) and amissile terminal guidance law f₃(z), wherein z represents a vectorcontaining fuzzy membership functions and guidance rules associated witheach of the missile guidance laws; b) a second sequence of instructionswhich, when executed by the processor, causes the processor to optimizethe missile launch guidance law f₁(z), the missile midcourse guidancelaw f₂(z) and the missile terminal guidance law f₃(z) by simultaneouslyminimizing the following set of equations:${Minmize}\mspace{14mu} \left\{ \begin{matrix}{{f_{1}(z)} = t_{f}} \\{{f_{2}(z)} = {\int_{0}^{t_{f}}{a_{M}^{2}{t}}}} \\{{f_{3} = {r\left( t_{f} \right)}},}\end{matrix} \right.$ wherein t_(f) represents missile interceptiontime, t represents time, r represents a distance between the missile anda target, and a_(M) represents a missile normal acceleration, andfurther |r(t_(f))|<r_(miss-allowed), where r_(miss-allowed) represents apre-selected allowable miss distance, wherein the minimization isperformed by a strength Pareto evolutionary defined by the followingsets of instructions: c) a third sequence of instructions which, whenexecuted by the processor, causes the processor to initialize a feasiblepopulation by generating an initial population and generating an emptyexternal Pareto-optimal set, the feasible population being selected tosatisfy a set of missile guidance constraints, wherein the set ofmissile guidance constraints includes the pre-selected allowable missdistance; d) a fourth sequence of instructions which, when executed bythe processor, causes the processor to search the feasible populationfor non-dominated individuals and copying the non-dominated individualsinto the external Pareto set; e) a fifth sequence of instructions which,when executed by the processor, causes the processor to search theexternal Pareto set for the non-dominated individuals and removing alldominated solutions from the external Pareto set, wherein if the numberof the individuals stored in the external Pareto set exceeds apre-specified maximum size, then the set is reduced by clustering; f) asixth sequence of instructions which, when executed by the processor,causes the processor to assign a strength to each individual in theexternal Pareto set, wherein the strength is proportional to the numberof individuals covered by that individual; g) a seventh sequence ofinstructions which, when executed by the processor, causes the processorto calculate a fitness of each individual in the population as the sumof the strengths of all external Pareto solutions which dominate thatindividual; h) an eighth sequence of instructions which, when executedby the processor, causes the processor to combine the population and theindividuals of the external Pareto set; i) a ninth sequence ofinstructions which, when executed by the processor, causes the processorto randomly select two individuals and comparing their respectivefitnesses; j) a tenth sequence of instructions which, when executed bythe processor, causes the processor to select the individual with thegreater fitness and copying the individual with the greater fitness to amating pool; k) an eleventh sequence of instructions which, whenexecuted by the processor, causes the processor to perform crossover andmutation operations to generate a new population; and l) a twelfthsequence of instructions which, when executed by the processor, causesthe processor to check for pre-selected stopping criteria and if apre-selected stopping criterion is satisfied, then ceasing optimizationand recording the optimal population, and if a pre-selected stoppingcriterion is not satisfied, then replacing the previous population withthe new population and returning to the fourth sequence of instructions,wherein the pre-selected stopping criteria includes a miss distance thatis less than the pre-selected allowable miss distance.
 11. The computersoftware product that includes a medium readable by a processor, themedium having stored thereon a set of instructions for generating anintegrated guidance law for aerodynamic missiles as recited in claim 10,further comprising a thirteenth sequence of instructions following theseventh sequence of instructions, which cause the processor to add apre-selected positive number to the sum of the strengths.
 12. Thecomputer software product that includes a medium readable by aprocessor, the medium having stored thereon a set of instructions forgenerating an integrated guidance law for aerodynamic missiles asrecited in claim 11, further comprising a fourteenth sequence ofinstructions which, when executed by the processor, causes the processorto establish an i-th membership function μ_(i) of an i-th objectivefunction of the Pareto-optimal set F_(i), wherein i is an integer, isdefined as: $\mu_{i} = \left\{ \begin{matrix}{1,} & {{F_{i} \leq F_{i}^{m\; i\; n}},} \\{\frac{F_{i}^{{ma}\; x} - F_{i}}{F_{i}^{{ma}\; x} - F_{i}^{m\; i\; n}},} & {{F_{i}^{m\; i\; n} < F_{i} < F_{i}^{{ma}\; x}},} \\{0,} & {F_{i\;} \geq {F_{i}^{{ma}\; x}.}}\end{matrix} \right.$ wherein f_(i) ^(max) and F_(i) ^(min) representthe maximum and minimum values of the i-th objective function,respectively.
 13. The computer software product that includes a mediumreadable by a processor, the medium having stored thereon a set ofinstructions for generating an integrated guidance law for aerodynamicmissiles as recited in claim 12, further comprising a fifteenth sequenceof instructions which, when executed by the processor, causes theprocessor to calculate a normalized membership function μ_(k) for eachnon-dominated solution k as:${\mu^{k} = \frac{\sum\limits_{i = 1}^{N_{obj}}\mu_{i}^{k}}{\sum\limits_{j = 1}^{M}{\sum\limits_{i = 1}^{N_{obj}}\mu_{i}^{j}}}},$wherein j is an integer, N_(obj) represents the number of objectives inthe multi-objective optimization, and M represents the number ofnon-dominated solutions.